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Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
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Bugeaud, Yann, Mignotte, Maurice and Siksek, Samir (2006) Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematics, Vol.163 (No.3). pp. 969-1018. doi:10.4007/annals.2006.163.969
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Official URL: http://www.jstor.org/stable/20159981
Abstract
This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science > Mathematics | ||||
| Journal or Publication Title: | Annals of Mathematics | ||||
| Publisher: | Mathematical Sciences Publishers | ||||
| ISSN: | 0003-486X | ||||
| Official Date: | May 2006 | ||||
| Dates: |
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| Volume: | Vol.163 | ||||
| Number: | No.3 | ||||
| Number of Pages: | 50 | ||||
| Page Range: | pp. 969-1018 | ||||
| DOI: | 10.4007/annals.2006.163.969 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Funder: | Sultan Qaboos University (Oman) | ||||
| Grant number: | IG/SCI/DOMS/02/06 |
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